# Surrounded and Covered

In the problem Surrounded and Covered, students use one- and two-dimensional measurement to solve problems involving area and perimeter. The mathematical topics that underlie this problem are the attributes of linear measurement, square measurement, two-dimensional geometry, perimeter, area, functions, fractal geometry, recursion, and infinity. The problem asks students to explore the relationship of area and perimeter in various problem situations. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students are presented with 12 squares that make a 2 by 6 rectangular patio. The students are asked to rearrange patio tiles to make rectangular figures that are different sizes (perimeters). They are also asked whether they can make a square patio from the 12 square tiles.

LEVEL A
In this level, students are presented with 12 squares that make a 2 by 6 rectangular patio. The students are asked to rearrange patio tiles to make rectangular figures with different perimeters. They are also asked whether they can make a square patio from the 12 square tiles.

Students will need to understand that they can count unit squares in order to find the area of a figure (3.MD.C.6).

LEVEL B
In this level, students are presented with irregular shapes on grid paper and are asked to determine the figure with the largest area and the largest perimeter.

Students will be solving real-world problems involving finding the area and perimeter of a figure (3.MD.D.8).

LEVEL C
In this level, students explain the results of changing the area of a figure. Students must make sense of why the price per square foot of sod must change at a different rate than linear scale factor when enlarging a geometric shape.

Students will use a ratio in the context of the relationship of length and width of a rectangle. The problem also involves students using unit pricing (6.RP.A.2, 6.RP.A.3b).

LEVEL D
In this level, the students explore the concepts of maximizing area given a fixed perimeter. The students grapple with identifying the quadrilateral that will produce the largest area. Finally, the students explore other polygons and are confronted with finding the geometric shape that will produce the largest area given a fixed-length boundary.

Students will solve real-world problems involving the area of two-dimensional objects including a rectangle, hexagon, and circle (7.G.B.4, 7.G.B.6).

LEVEL E
In this level, students explore area and perimeter in terms of fractal geometry.  Students are introduced to Sierpinski’s triangle and are asked to determine the area and perimeter of the fractal at each stage of the fractal’s development, including finding the area and perimeter for the nth stage.

This level addresses Common Core standard F-BF-A.1.a by asking students to describe the relationship between the area and perimeter of a Sierpinski triangle for any given stage using a function rule.

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